Smooth Cutoff Research Articles

Synchrotron Radiation as the Source of Gamma‐Ray Burst Spectra

We investigate synchrotron emission models as the source of gamma-ray burst spectra. We show that including the possibility for synchrotron self-absorption, a ``smooth cutoff'' to the electron energy distribution, and an anisotropic distribution for the electron pitch angles produces a whole range of low energy spectral behavior. In addition, we show that the procedure of spectral fitting to GRB data over a finite bandwidth can introduce a spurious correlation between spectral parameters - in particular, the value of the peak of the nu F_nu spectrum, E_p, and the low energy photon spectral index alpha (the lower E_p is, the lower (softer) the fitted value of alpha will be). From this correlation and knowledge of the E_p distribution, we show how to derive the expected distribution of alpha. We show that optically thin synchrotron models with an isotropic electron pitch angle distribution can explain the distribution of alpha below alpha=-2/3. This agreement is achieved if we relax the unrealistic assumption of the presence of a sharp low energy cutoff in the spectrum of accelerated electrons, and allow for a more gradual break. We show that this low energy portion of the electron spectrum can be at most flat. We also show that optically thin synchrotron models with an anisotropic electron pitch angle distribution can explain all bursts with -2/3 < alpha <= 0$. The very few bursts with low energy spectral indices that fall above alpha=0 may be due the presence of a the synchrotron self-absorption frequency entering the lower end of the BATSE window. Our results also predict a particular relationship between alpha and E_p during the temporal evolution of a GRB. We give examples of spectral evolution in GRBs and discuss how the behavior are consistent with the above models.

Glueballs in a Hamiltonian light-front approach to pure-glue QCD

We calculate a renormalized Hamiltonian for pure-glue QCD and diagonalize it. The renormalization procedure is designed to produce a Hamiltonian that will yield physical states that rapidly converge in an expansion in free-particle Fock-space sectors. To make this possible, we use light-front field theory to isolate vacuum effects, and we place a smooth cutoff on the Hamiltonian to force its free-state matrix elements to quickly decrease as the difference of the free masses of the states increases. The cutoff violates a number of physical principles of light-front pure-glue QCD, including Lorentz covariance and gauge covariance. This means that the operators in the Hamiltonian are not required to respect these physical principles. However, by requiring the Hamiltonian to produce cutoff-independent physical quantities and by requiring it to respect the unviolated physical principles of pure-glue QCD, we are able to derive recursion relations that define the Hamiltonian to all orders in perturbation theory in terms of the running coupling. We approximate all physical states as two-gluon states, and use our recursion relations to calculate to second order the part of the Hamiltonian that is required to compute the spectrum. We diagonalize the Hamiltonian using basis-function expansions for the gluons' color, spin, and momentum degrees of freedom. We examine the sensitivity of our results to the cutoff and use them to analyze the nonperturbative scale dependence of the coupling. We investigate the effect of the dynamical rotational symmetry of light-front field theory on the rotational degeneracies of the spectrum and compare the spectrum to recent lattice results. Finally, we examine our wave functions and analyze the various sources of error in our calculation.

Convergence of derivative expansions of the renormalization group

We investigate the convergence of the derivative expansion of the exact renormalization group, by using it to compute the beta function of scalar field theory. We show that the derivative expansion of the Polchinski flow equation converges at one loop for certain fast falling smooth cutoffs. The derivative expansion of the Legendre flow equation trivially converges at one loop, but also at two loops: slowly with sharp cutoff (as a momentum-scale expansion), and rapidly in the case of a smooth exponential cutoff. Finally, we show that the two loop contributions to certain higher derivative operators (not involved in beta) have divergent momentum-scale expansions for sharp cutoff, but the smooth exponential cutoff gives convergent derivative expansions for all such operators with any number of derivatives.

Application of renormalization-group techniques to a homogeneous Bose gas at finite temperature

A homogeneous Bose gas is investigated at finite temperature using renormalization group techniques. A non--perturbative flow equation for the effective potential is derived using sharp and smooth cutoff functions. Numerical solutions of these equations show that the system undergoes a second order phase transition in accordance with universality arguments. We obtain the critical exponent $\nu =0.73$ to leading order in the derivative expansion.

Multiscale Formalism for Correlation Functions of Fermions. Infrared Analysis of the Tridimensional Gross–Neveu Model

We present a multiscale formalism for fermionic systems (with a smooth UV cutoff ) establishing a trivial link between the correlation functions and the effective potential flow, and study the k-point truncated functions of the tridimensional Gross–Neveu model. A new efficient method is used to bound these correlation functions and show polynomial tree decay for long distances. We are guided by a block lattice mechanism with a property of orthogonality between terms in different scales, which leads to simple formulas for the correlations.

Regularization and renormalization in effective field theories of the nucleon-nucleon interaction

Some form of nonperturbative regularization is necessary if effective field theory treatments of the NN interaction are to yield finite answers. We discuss various regularization schemes used in the literature. Two of these methods involve formally iterating the divergent interaction and then regularizing and renormalizing the resultant amplitude. Either a (sharp or smooth) cutoff can be introduced, or dimensional regularization can be applied. We show that these two methods yield different results after renormalization. Furthermore, if a cutoff is used, the NN phase shift data cannot be reproduced if the cutoff is taken to infinity. We also argue that the assumptions which allow the use of dimensional regularization in perturbative EFT calculations are violated in this problem. Another possibility is to introduce a regulator into the potential before iteration and then keep the cutoff parameter finite. We argue that this does not lead to a systematically-improvable NN interaction.

Averages over curves with torsion

We establish L p Sobolev mapping properties for averages over certain curves in R 3 , which improve upon the estimates obtained by L 2 L 1 interpolation. Let T be the operator given by convolution in R 3 against a smooth cutoff of arclength measure on the helix (t) = cost,sint,t � , Tf(x) = Z f(x1 − cost, x2 − sint, x3 −t)�(t)dt. For 1 < p < ∞, let H s,p (R 3 ) denote the nonhomogeneous Sobolev space consisting of functions in L p (R 3 ) whose fractional derivative of orders also lies in L p (R 3 ). We consider the following question:

The derivative expansion of the renormalization group

By writing the flow equations for the continuum Legendre effective action (a.k.a. Helmholtz free energy) with respect to a particular form of smooth cutoff, and performing a derivative expansion up to some maximum order, a set of differential equations are obtained which at FPs (Fixed Points) reduce to non-linear eigenvalue equations for the anomalous scaling dimension $\eta$. Illustrating this by expanding (single component) scalar field theory, in two, three and four dimensions, up to second order in derivatives, we show that the method is a powerful and robust means of discovering and quantifying non-perturbative continuum limits (continuous phase transitions).

Derivative expansion of the exact renormalization group

The functional flow equations for the Legendre effective action, with respect to changes in a smooth cutoff, are approximited by a derivative expansion; no other approximation is made. This results in a set of coupled non-linear differential equations. The corresponding differential equations for a fixed point action have at most a countable number of solutions that are well defined for all values of the field. We apply the technique to the fixed points of one-component real scalar field theory in three dimensions. Only two non-singular solutions are found; the gaussian fixed point and an approximation to the Wilson fixed point. The latter is used to compute critical exponents, by carrying the approximation to second order. The results appear to converge rapidly.

Theoretical calculation of thermodynamic data for bcc binary alloys with the embedded-atom method.

The dilute-limit heats of solution for all binary alloys of six bcc transition metals (V, Nb, Ta, Mo, W, and Fe) have been calculated with the analytic embedded-atom model for bcc pure metals by Johnson and Oh. Cubic equations are proposed and used for providing a smooth cutoff for the calculated potential and electron density functions between second- and third-nearest neighbors in the calculation. The heats of formation for all of the binary alloys of these six bcc metals for the whole compositional range and the intermetallic compounds ${\mathit{A}}_{3}$B, AB, and ${\mathit{AB}}_{3}$ are also calculated. The dilute-limit heats of solution are generally in agreement with available experimental values except for Ta in W and W in Ta. The heats of formation agree well with available experimental data, ab initio calculations by Colinet, Beesound, and Pasturrel, and thermodynamic calculations with the Miedema model for Mo-Ta, Mo-Nb, and Fe-V. The heats of formation are in good agreement with thermodynamic calculations with the Miedema model for W-V, Nb-V, Ta-V, Fe-W, Mo-W, Ta-Nb, Fe-Nb, Fe-Ta, and Mo-V alloy systems. There are, however, significant differences between the heats of formation for the present work and calculations with the Miedema model for the Ta-W, Nb-W, and Fe-Mo alloy systems.

Local cosine transform ? a method for the reduction of the blocking effect in JPEG

This paper presents the local cosine transform (LCT) as a new method for the reduction and smoothing of the blocking effect that appears at low bit rates in image coding algorithms based on the discrete cosine transform (DCT). In particular, the blocking effect appears in the JPEG baseline sequential algorithm. Two types of LCT were developed: LCT-IV is based on the DCT type IV, and LCT-II is based on DCT type II, which is known as the standard DCT. At the encoder side the image is first divided into small blocks of pixels. Both types of LCT have basis functions that overlap adjacent blocks. Prior to the DCT coding algorithm a preprocessing phase in which the image is multiplied by smooth cutoff functions (or bells) that overlap adjacent blocks is applied. This is implemented by folding the overlapping parts of the bells back into the original blocks, and thus it permits the DCT algorithm to operate on the resulting blocks. At the decoder side the inverse LCT is performed by unfolding the samples back to produce the overlapped bells. The purpose of the multiplication by the bell is to reduce the gaps and inaccuracies that may be introduced by the encoder during the quantization step. LCT-IV and LCT-II were applied on images as a preprocessing phase followed by the JPEG baseline sequential compression algorithm. For LCT-IV, the DCT type IV replaced the standard DCT as the kernel of the transform coding. In both cases, for the same low bit rates the blocking effect was smoothed and reduced while the image quality in terms of mean-square error became better. Subjective tests performed on a group of observers also confirm these results. Thus the LCT can be added as an optional step for improving the quality of existing DCT (JPEG) encoders. Advantages over other methods that attempt to reduce the blocking effect due to quantization are also described.

A new interpretation of dimensional regularization

A frequently used method to regulate the Nambu-Jona-Lasinio model is to attach a sharp covariant Euclidean cutoff. If one replaces the sharp cutoff by a smooth cutoff, the integrals of dimensional regularization are found. The Dirac algebra required by such a smooth cutoff regularization method cannot be represented by matrices, however. Therefore we suggest an operator representation of the Dirac algebra. In particular, an explicit operator representation corresponding to the Dirac matrix γ 5 is given. The anomaly of the axial vector current is derived from the operator algebra. The formalism is applied to regularize the Nambu-Jona-Lasinio model. The decay width of the pion is investigated both in the sharp cutoff and in the dimensional regularization.

Radiative capture in the light heavy-ion systems 6Li+ 6Li and 6Li+ 7Li at Ec.m. = 1–8 MeV

The heavy-ion radiative-capture reactions 6Li( 6Li, γ) 12C and 7Li( 6Li, γ) 13C have been studied. Utilizing a large-volume NaI(Tl) γ-spectrometer, the γ-transitions to the ground state and the three lowest excited states of 12C in the former case and the transitions γ 0, γ 1, γ 2 3 , γ 4 and γ 5 6 7 in the latter case could be resolved. The evaluation of absolute cross sections which range from 0.3 to 210 nb/sr relies on the thoroughly investigated response function of the spectrometer. Excitation functions at 125° as well as three-point angular distributions at several energies were measured. The results are compared with statistical model calculations. The latter make use of a smooth angular momentum cutoff in the entrance channel, isospin mixing in the compound nucleus, and giant resonances built upon excited states for γ-strenght distributions. All data are described by these model calculations. In addition to these statistical features a broad resonance ( gL ∼ 0.8 MeV) is observed for th γ 1-transition in 6Li( 6Li, γ) 12C at E x ∼ 30.3 MeV.

Application of the statistical model to the light heavy-ion system6Li +6Li

Reactions between6Li and6Li were studied in the laboratory energy range of 2–16 MeV. Angular distributions and excitation functions were measured for various proton, deuteron, triton andα-channels. In addition excitation functions were obtained for five γ-transitions in10B and the ground-state transitions in both7Li and7Be. Detailed statistical model calculations were performed under various assumptions concerning the formation of the compound nucleus. It is concluded that a smooth cutoff of the entrance channel angular momenta yields the only consistent description for both angular distributions and excitation functions. Only a few proton channels are dominated by compound nucleus decay. All other channels reflect the complex structure of6Li which causes a variety of direct reaction modes.

Symanzik's improved actions from the viewpoint of the renormalization group

We investigate Symanzik's improvement program in a four-dimensional Euclidean scalar field theory with smooth momentum space cutoff. We use Wilson's renormalization group transformation to define the improved actions as a sequence of initial data for the effective action at the fundamental cutoff. This leads to a sequence of solutions to the renormalization group equation. We define the parameters of the improved actions implicitly by conditions on the effective action at a renormalization scale. The improved actions are close approximations to the continuum effective action. We prove their existence to every order of improvement and to every order of renormalized perturbation theory.

The space distribution of faint quasars from the CFHT survey

Basic data are presented for 163 QSOs brighter than B = 20.5 selected from 1000 A/mm CFHT blue grens observations. The data are used to investigate the form of the evolution of QSOs in order to obtain their luminosity functions, and to predict their magnitude and density distributions to allow comparison with the results of other surveys. Less than 5 percent of the QSOs have z greater than 2.5 despite the ease of detecting them up to z = 3.4, so high-redshift QSOs are rare. The surface density of QSOs brighter than m = 20.5 is 30/sq deg. While the luminosity-dependent density evolution models of Schmidt and Green (1983) are found to fit the data well, a smooth cutoff to the evolution at high redshifts is proposed which implies a density maximum at z of roughly 1.7. The clustering properties of the sample QSOs are discussed and it is shown, in agreement with previous studies, that while clustering is not a general phenomenon among QSOs, significant density enhancements apparently do exist. 26 references.

Radius anomaly in the diffraction model for heavy-ion elastic scattering

The elastic scattering of heavy ions, $^{20}\mathrm{Ne}$ on $^{208}\mathrm{Pb}$, $^{20}\mathrm{Ne}$ on $^{235}\mathrm{U}$, $^{84}\mathrm{Kr}$ on $^{208}\mathrm{Pb}$, and $^{84}\mathrm{Kr}$ on $^{232}\mathrm{Th}$, is examined within the framework of Frahn's diffraction model. An analysis of the experiment using the "quarter point recipe" of the expected Fresnel cross sections yields a larger radius for $^{208}\mathrm{Pb}$ than the radii for $^{235}\mathrm{U}$ and $^{232}\mathrm{Th}$. It is shown that inclusion of the nuclear deformation in the model removes the above anomaly in the radii, and the assumption of smooth cutoff of the angular momentum simultaneously leads to a better fit to elastic scattering data, compared to those obtained by the earlier workers on the assumption of sharp cutoff.[NUCLEAR REACTIONS Elastic scattering, $^{20}\mathrm{Ne}$+$^{208}\mathrm{Pb}$ (161.2 MeV), $^{20}\mathrm{Ne}$+$^{235}\mathrm{U}$ (175 MeV), $^{84}\mathrm{Kr}$+$^{208}\mathrm{Pb}$ (500 MeV), $^{84}\mathrm{Kr}$+$^{232}\mathrm{Th}$ (500 MeV), diffraction model, nuclear deformation.]

Direct and compound nuclear contributions to the 13C( 6Li, t) reaction at Elab = 25 MeV

The reaction 13C( 6Li, t) 16O has been studied in the incident energy range 24–26 MeV. Complete angular distributions have been measured at E 6Li , = 25 MeV in the angular range θ lab = 8°–172°, with the reaction 6Li( 13C, t) 6O being used for the backward angle measurements. Cross sections for evaporation residues from the fusion of the 6Li + 13C system have been measured in the incident 6Li energy range 9.2–35.1 MeV. Compound nuclear contributions to the transfer cross sections have been calculated using the Hauser-Feshbach statistical theory with the assumption that the compound-nucleus formation cross section is equal to the measured fusion cross section. By comparison of the compound nuclear calculations with backward angle data it is found that the sharp cutoff approximation commonly used to represent the initial angular momentum distribution of the compound nucleus is not adequate for the 13C( 6Li, t) 16O reaction. Good fits to the backward angle data can be obtained by using a smooth cutoff approximation. The forward angle cross sections have been compared with exact finite-range distorted-wave Born approximation calculations to extract transferred angular momenta and spectroscopic strengths. The present results differ from those of an earlier study. These differences are due to the inclusion of forward angle data in the present study.

Continuum strong-coupling expansion for quantum electrodynamics

We derive from the path integral a continuum strong-coupling expansion for QED in $d$-dimensional Euclidean space-time. It is a double expansion in the fermion and boson kinetic energy (inverse free propagators), which leads to a double power series for the Green's functions of the cutoff theory in terms of $\frac{1}{{e}^{2}}$ and $\frac{{\ensuremath{\Lambda}}^{2}}{{M}^{2}}$. $\ensuremath{\Lambda}$ is a smooth cutoff in Euclidean momentum space, and $M$ is an infrared regulator mass for the photons needed to define the local part of the path integral. We demonstrate how dimensional continuation is necessary to control the broken gauge invariance of the cutoff theory. Restricting to $d=2$ (the Schwinger model) we show how to remove the cutoff using Pad\'e approximants. We find some evidence that as $\frac{{\ensuremath{\Lambda}}^{2}}{{M}^{2}}\ensuremath{\rightarrow}\ensuremath{\infty}$ gauge invariance is restored and we calculate the vector-meson mass, keeping the first three terms in the expansion in powers of the bare photon inverse propagator.

Substrate strain induced interaction of adatoms on W (110)

The interaction of adatoms due to elastic strains created in an elastically isotropic substrate is investigated. For cases where the adatoms occupy sites with low symmetry, an angular dependent interaction results which falls off as s−3 at large distances. An exact expression is given for the long range interaction in terms of an anisotropy parameter of the force dipole tensor. The short range interaction is calculated by introducing a smooth cutoff. Interactions of adatoms on near neighbour sites on W (110) are given.